Math Is Fun Forum

sachik

Member

Registered: 2008-02-28

Posts: 9

a few word problems i dont know how to solve

1) a pizza show makes $1.50 on each small pizza and $2.15 on each large pizza. On a typical friday, it sells between 70 and 90 small pizzas and between 100 and 140 large pizzas. The shop can make no more than 210 pizzas a day. How many of each size of pizza must be sold in order to maximize profit?

2) The area of a rectangle is 36². The perimeter of the rectangle is 36 in. Find the dimensions of the rectangle to the nearest hundredth of an inch.

3) The product of two positive numbers is 96 and the sum of their squares is 208. What are the two numbers.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: a few word problems i dont know how to solve

1) The shop has to make at least 70 small pizzas and 100 large ones, so after that it has 40 "free" pizzas left.
To maximise profit, it needs to make as many of those big as it can. Luckily, making all of them big won't break the limit of 140 big pizzas, so it does that.

Then, its profit is 70*$1.50 + 140*$2.15 = $406.

2) Calling x and y the dimensions of the rectangle, you have:

xy = 36
2(x+y) = 36

Therefore y = 36-x, and so x(36-x) = 36².
Arrange this into a quadratic equation and solve for x, then use that answer in one of the original equations to find y.

3) is done in a similar way to 2).

Thanks ganesh

Last edited by mathsyperson (2008-05-18 01:01:23)

---

Why did the vector cross the road?
It wanted to be normal.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 46,341

Re: a few word problems i dont know how to solve

sachik,
There's some mistake in the second question. If the perimeter is 36 in, the sum of the length and the breadth is 18 in. This being the case, the area can never be 36². I guess the question should read : the area is 36 in².
Assuming so, you get a quadratic equation
x²-18x+36=0
Solve for x using the quadratic equation formula, This is one of the dimensions. The other one is 18-x.

For the third question,
x²+y²=208
xy=96
Therefore, 2xy=192
Therefore, x²+y²-2xy=208-192=16
(x-y)²=16; x-y=4

Since the product of the numbers is 96,
x(x-4) = 96.

Therefore, the numbers are 8 and 12.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

sachik

Member

Registered: 2008-02-28

Posts: 9

Re: a few word problems i dont know how to solve

Hi ganesh,

xy=96
Therefore, 2xy=192
Therefore, x²+y²-2xy=208-192=16
(x-y)²=16; x-y=4

Since the product of the numbers is 96,
x(x-4) = 96.

Therefore, the numbers are 8 and 12.

could you explain this part a little more:
Since the product of the numbers is 96,
x(x-4) = 96.

Therefore, the numbers are 8 and 12.

how and why did you do that?

thank you

Last edited by sachik (2008-05-22 19:31:22)

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 46,341

Re: a few word problems i dont know how to solve

We know xy=96 and x-y=4.
x-y=4 =>-y=4-x =>y=x-4
Therefore x(x-4)=96

Actually, it was a little calculation I performed in my mind and said the numbers are 8 and 12.

This is how it is done.
x(x-4)=96 or
x²-4x=96 or x²-4x-96=0.
Solving the quadratic equation for x, you get x=8,12.

---

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.